Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 218 / 6 / c / ii

Write $\Phi \alpha ={\sum }_i{\alpha }_i\varphi (X_i)$. By the definition of the kernel matrix,andThe irrelevant positive factor $1/n$ gives exactly the stated constrained optimization.

Let $K{v}_1={\lambda }_1{v}_1$, where ${\lambda }_1>0$ is the largest eigenvalue and $\Vert v_1{\Vert }_2=1$. The generalized Rayleigh quotient is maximized byup to sign and addition of a vector in $\ker K$, which does not change $\hat{u}=\Phi \hat{\alpha }$. Repeated leading eigenvectors give further principal directions.